The Pre-schwarzian Derivatives And Schwarzian Derivatives Of Harmonic Mappings

In this thesis,we introduce a definition of the pre-Schwarzian and Schwarzain derivatives of locally univalent harmonic mapping on a simple connected domain in com-plex plane.We discuss the analytic and harmonic conditions of the pre-Schwarzian and Schwarzain derivatives of the harmonic mapping.we also study related Schwarzian deriva-tive norm theory.In the early 1920s,differential geometric scholars began to study the harmonic func-tion in detail because of the close connection between harmonic mapping and minimal surfaces.Chuaqui,Duren and Osgood proposed the concept of Schwarzian derivative on planar harmonic function according to the differential geometry of minimal surfaces.In terms of the Weierstrass-Enneper formulas,the locally univalent sense-preserving har-monic functions can lift into a regular minimal surface.Using the Riemann metric on the minimal surface as the conformal metric,they define the Schwarzian derivative of the plane harmonic mapping.But their definition has some limitations:harmonic mapping lifts to a mapping onto a minimal surface defined by conformal parameters if and only if the dilatation equals the square of an analytic function.In order to avoid the Schwarzian derivative on the requirements of the dilatation,using the energy density as the conformal metric density,we introduce a definition of the pre-Schwarzian and Schwarzain derivatives of locally univalent harmonic mapping on a simple connected domain in complex plane.We also discuss some properties of the pre-Schwarzian and Schwarzain derivatives.This paper is composed of four chapters.In chapter one,we briefly introduce,the background of harmonic function theory,some definitions and notations in this thesis.The Schwarzian derivative of a locally univalent analytic function is analytic and it is easy to introduce its pre-Schwarzian derivative is also analytic.Harmonic function is the extension of analytic function,so it is necessary to discuss the analytic and harmonic conditions of the pre-Schwarzian derivative and Schwarzian derivative of the harmonic function.In chapter two,we discuss this problem.In 1949,Nehari discovered the relation between the Schwarzian derivative of analytic function and univalent of analytic function.Then,using the norm of Schwarzian derivative,Some scholars gave some conditions of univalent of analytic function.In this process it need to estimate the norm of Schwarzian derivative,but it is not a simple problem.In chapter three,we study this problem and give some estimates of norm of the Schwarzain derivatives of convex harmonic mapping in the unit disk and discuss the boundedness of norm of the Schwarzain derivatives.If some univalent analytic functions satisfy the univalent criterions of Nehari,they called Nehari function families.Nehari function families have some nice property,so in chapter four,we define an Similar subclass of harmonic function.We give some estimates of the pre-Schwarzian and the conformal metric density.